The Multiple Game

topIn this week’s post we look at a game for practicing recognition of multiples of a number. It can be used in class by a teacher or it might be a good game to play with your kid by way of some math practice. The game prepares a student for multiplication by practicing how to recognize which numbers are multiples of five. The numbers 5, 10, 15, and so on generate scores when they come up in the course of the game. There are multiple versions of the game (for other numbers besides five), but the demonstrations are based on recognizing multiples of five. The game is a little like dominos and it’s a little like a crossword puzzle.

The board is 10×10 and each player gets a collection of numbers they can play, one per grid, according to the rules below. There is also a “number of the day”. Multiples of this special number permit a player to score. For the examples below, the number of the day is five. The set of four ones, four twos, four threes, and four fours are also chosen for a game where the number of the day is five. This game is intended for students that are learning their basic multiplication facts.

A first player is determined randomly; you might flip a coin. The red player puts numbers from his supply of playing pieces on the board in red, the blue player in blue.

  1. Players alternate playing numbers from their supply, crossing off numbers when they are used. Alternating play is suspended after a player scores and the scoring player is permitted an additional move.
  2. Except for the first number, which may be played anywhere, numbers may only be played adjacent to other numbers, horizontally, vertically, or diagonally.When the numbers in a horizontal, vertical, or diagonal line are a multiple of the number of the day, and have numbers of both colors on the line, a line is drawn through those numbers and their total is added to the player’s score.
  3. If a player does not notice they have scored and does not cross off their scoring line before the next move is made, they lose that score.
  4. When a player scores, they must play again immediately. A player may play several times in a row as long as they score each time to earn the next play.
  5. A crossed-off line may not be extended by playing numbers adjacent to its ends: those squares are available for play because of other adjacencies.
  6. If a player runs out of numbers, the other player then plays all
    of their remaining numbers. Play continues until neither player has
    any remaining numbers.
  7. The winner is the player with the highest score. If there is a
    tie in scoring, the first player to run out of numbers is the winner.

It is now time to look at an example of play. The red player gets the first move and plays a 1. The blue player plays a 4 and scores 5 points.

Board1

The blue player must move again and plays a 1. The red player then also plays a one, getting a diagonal score of 5.

Board2

The red player must play again and plays a 4, getting another score of 5.

Board3

The red player must play again and plays a 3. The blue player plays a 2 to get a vertical score of 10 and a one to get a horizontal score of 5. The blue player is way ahead.

Board4

The blue player now plays a 4 to pick up a vertical score of 10, making her total 30. The blue player finishes her turn by putting down a 3. She does not notice that the 3 completes a diagonal score of 5, and so loses that score. This leads to a loud groan after the red player’s next move.

Board5

The red player puts down a 1 and gets a horizontal score of 10.

Board6

The red player now puts down a 2, getting a vertical score of 5. The score is now red 25 and blue 30. The red player has ten pieces left, the blue player only has eight, and the red player is about to play again.

Board7

Occupy Math has a confession to make. The illustration of missing a score, leading to the blue player’s groan, is Occupy Math’s own error. Taking that score would have involved going back and doing several drawings over again — until Occupy Math noticed he could illustrate the rule about missing a score.

Tactics, strategy, and game balance

It is not clear if the first or second player of the multiple game has an advantage. Certainly going second creates an early advantage, but this sets up an end-of-game where the first player may have several unanswered plays in a row. The strategy, setting up future situations for yourself without giving too much to the other player, is quite complex.

One important tactical point is this — if you run out of one sort of play, it is much easier for your opponent to create rows where he can score and you cannot. If we have a 4 4 3 row, then a 4 is needed to get a score of 15. The four is needed to make the play. Also notice that making a multiple of the number of the day in just one color is a clever move. You cannot score immediately, but if your opponent plays at either end of the line, you get a big score (if, of course, you have the right piece left).

The example game is done assuming the number of the day is 5. It is not a good idea to have a number of the day smaller than five — the game gets too restricted. There should be a equal number of pieces of each type from 1 up to one less than the number of the day. So, if the number of the day is 6, the pieces each player has should be three each of 1, 2, 3, 4, and 5. If the number of the day is 7, use three each of 1, 2, 3, 4, 5, and 6. If the number of the day is 8, then two each of 1, 2, 3, 4, 5, 6, and 7, and so on.

Alternative rules

Here are some alternative rules for the game.

  • No diagonals. This makes the game simpler.
  • Instead of scoring based on a number of the day, use the primes 2, 3, 5, 7, 11, and 13. The first player to make a multiple of a prime gets the score and then the prime is crossed off. This requires somewhat more advanced mathematical skills.
  • Roll two six-sided dice to get the number for which multiples of that number score. As soon as a score is made, roll again. This makes the game more of a game of chance.
  • Use only the numbers 1, 2, and 3 as playing pieces. This increases the need for tactical play by reducing the number of instant score moves.

Concluding Thoughts

A good exercise for a group is to try and determine, for the original game with 5 as the number of the day, if the group thinks there is a first player or a second player advantage. Presenting evidence and conclusions for this supposition is itself a nice piece of mathematics practice. This can also be done again for other numbers of the day and the conclusion might be different.

It might also be good to try to construct variations of your own on this game. The key features are the crossing-off mechanic and needing both colors in a line to score. These work together to keep the game from being trivial. Other mechanics might do a better job. If you find some, Occupy Math would love to hear about it in the comments.

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics

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